F-theta lens

Abstract: An f-theta lens includes three lenses in two groups. The first lens group from the entrance pupil side consists of a first lens having a negative or positive refracting power. The second lens group consists of a second lens having a positive refracting power and a third lens having a negative refracting power with the second and third lenses being cemented together. The second lens group has a positive refracting power as a whole. The following formulae (1) to (4) are satisfied. 12≦ν d2 -ν d3 -0.035≦Σ( i /ν di )≦0.005 0.7≦ 23 ≦2.0 0.03≦d 12 ≦0.20 wherein  i represents the power of i-th lens as numbered from the entrance pupil side standardized on the basis of the power of the whole system,  23 represents the power of the second lens group standardized on the basis of the power of the whole system, ν di represents the Abbe's number for the sodium d-line of i-th lens as numbered from the entrance pupil side and d 12 represents the surface separation between the first and second lens groups standardized on the basis of the focal length of the whole system. The lens is designed to correct the color aberration, over a wide wavelength range with high accuracy. Use: scanning optical system for color image recording.

Description: BACKGROUND OF THE INVENTION Field of the Invention This invention relates to an f lens for a scanning optical system, and more particularly to an f lens suitable for use in a scanning optical system in which a plurality of laser beams having different wavelengths are used. Description of the Prior Art Recently there has been put into wide use various instruments such as a laser beam printer in which a laser beam scanning optical system is used. In such an optical system, an f lens is used to cause a laser beam, which is deflected by a rotary polygonal mirror or the like, to scan the surface of a medium at a constant speed. The f lens is a lens which satisfies a condition that Y=f wherein  represents the incident angle of the laser beam to the lens, f represents the focal length of the lens and Y represents the height of the image formed. Conventionally the f lens is designed on the assumption that a monochromatic or single wavelength laser beam is used and accordingly the color aberration causes no problem and correction of the color aberration has been unnecessary heretofore. However, for instance, in a laser recording system in which color images are recorded on silver salt film by use of a plurality of laser beams having different wavelengths, e.g., red, green and blue laser beams, the problem of color aberration must be taken into account. As is well known in the art, the color aberration includes the axial color aberration and the lateral color aberration. The axial color aberration shifts the position of the image in the direction of the optical axis depending on the wavelength and practically does not cause a serious problem since the focal depth of the laser beam is sufficiently large. However, the lateral color aberration causes the position of the image to shift in the direction perpendicular to the optical axis depending on the wavelength, and accordingly, the beam is shifted in the scanning direction by different distance depending on the wavelength, which makes it impossible to precisely superimpose the spots of the respective colors one on another and adversely affects the quality of images. Though this problem may be overcome by operating the laser beams of the respective colors in different timings, this approach requires complicated control and is not suitable for high-speed scanning. In Japanese Patent Publication No. 61(1986)-51289, there is disclosed a technique for correcting lateral color aberration. However this technique can correct lateral color aberration only by a wavelength of 20nm and is originally developed to correct color aberration due to fluctuation in wavelength of a semiconductor laser. Accordingly it cannot be applied to a multiple wavelength scanning optical system for color image recording. An f lens disclosed in Japanese Unexamined Patent Publication No. 62(1987)-262812 can correct the lateral color aberration for two or more laser beams with a high accuracy under predetermined conditions. However the f lens comprises four elements in three lens groups. When the number of lens elements is large, the manufacturing cost increases, and at the same time, errors in processing the lens elements and assembly of the same can accumulate. Further in order to satisfy the disclosed conditions, the material for the lens elements must be selected within a narrow range. SUMMARY OF THE INVENTION In view of the foregoing observations and description, the primary object of the present invention is to provide an f lens which can correct the color aberration with a high accuracy over a wide wavelength range, and consists of a small number of lens elements to permit reduction of the size of the optical system, the manufacturing cost and the error in assembly. In accordance with the present invention, there is provided an f lens comprising a first lens group which consists of a first lens having a negative or positive refracting power, and a second lens group which consists of a second lens having a positive refracting power and a third lens having a negative refracting power, the second and third lenses being cemented together, the second lens group having a positive refracting power as a whole, and the first and second lens groups being arranged in this order from the entrance pupil (deflecting surface) side and satisfying the following formulae (1) to (4), 12≦ν d2 -ν d3 -0.035≦Σ( i /ν di )≦0.005 0.7≦ 23 ≦2.0 0.03≦d 12 ≦0.20 wherein  i represents the power of i-th lens as numbered from the entrance pupil side standardized on the basis of the power of the whole system,  23 represents the power of the second lens group standardized on the basis of the power of the whole system, ν di represents the Abbe's number for the sodium d-line of i-th lens as numbered from the entrance pupil side and d 12 represents the surface separation between the first and second lens groups standardized on the basis of the focal length of the whole system. The lens elements of the f lens in accordance with the present invention need not be limited to spherical lenses but may be aspheric lenses. The f lens in accordance with the invention comprises only three lens elements in two lens groups and simple in structure. Further when the formulae (1) to (4) are satisfied, the color aberration can be corrected over a wide wavelength range. The formula (1) limits the relation between the Abbe's numbers of the second and third lenses of the second lens group, which is a cemented lens. The difference between the Abbe's numbers governs the amount of color aberration generated at the interface between the second and third lenses. When the difference between the Abbe's numbers of the second and third lenses is smaller than 12, the negative color aberration generated at the interface is reduced and the positive color aberration increases as a whole. The formula (2) relates to correction of the axial color aberration in paraxial theory and in order to correct the axial color aberration, the value of Σ( i /ν di ) must be substantially 0. Further in order to correct the lateral color aberration, the axial color aberration must be small to a certain extent. In accordance with the first aspect of the present invention, the relation between the power of each lens and the Abbe's number thereof is limited by the formula (2) in order to correct the lateral color aberration. When the value of Σ( i /ν di ) is smaller than the lower limit, the negative lateral color aberration increases and when the value of Σ( i /ν di ) is larger than the upper limit, the positive lateral color aberration increases. The formula (3) relates to the Petzval sum and limits the amount of curvature of field. That is, when the power of the second lens group standardized on the basis of the power of the whole system is larger than the upper limit, the f characteristic becomes too large in the positive direction to correct, and at the same time, the amount of curvature of field in the three-dimensional meridional image surface becomes too large in the negative direction to correct. On the other hand, when it is smaller than the lower limit, the amount of curvature of field in the three-dimensional meridional image surface becomes too large in the positive direction to correct. Further, since the f characteristic depends on the distance from the entrance pupil, the position of the first and second lens groups relative to the entrance pupil must be limited in order to correct the f characteristic. However since the position of the first lens group is more restricted by the system in which the f lens is employed and accordingly it is preferable to limit the position of the second lens system. That is, the formula (4) limits the position of the second lens group by limiting the distance between the first and second lens groups. When the distance between the first and second lens groups is smaller than the lower limit, the f characteristic becomes too large in the positive direction to correct, and when it is larger than the upper limit, the size of the second lens group increases, which adds to the manufacturing cost of the f lens. BRIEF DESCRIPTION OF THE DRAWINGS Figure 1A is a view showing the arrangement of the lens elements in an f lens of a first embodiment in accordance with the first aspect of the present invention, Figure 1B is a graph showing the lateral color aberration of the f lens shown in Figure 1A for three wavelengths W1, W2 and W3, Figure 1C is a graph showing the astigmatic field curves (curvature of field) of the f lens shown in Figure 1A for the three wavelengths W1, W2 and W3, Figure 1D is a graph showing the linearity (f characteristic) of the f lens shown in Figure 1A for the three wavelengths W1, W2 and W3, Figure 2A is a view showing the arrangement of the lens elements in an f lens of a second embodiment in accordance with the first aspect of the present invention, Figure 2B is a graph showing the lateral color aberration of the f lens shown in Figure 2A for three wavelengths W1, W2 and W3, Figure 2C is a graph showing the astigmatic field curves (curvature of field) of the f lens shown in Figure 2A for the three wavelengths W1, W2 and W3, Figure 2D is a graph showing the linearity (f characteristic) of the f lens shown in Figure 2A for the three wavelengths W1, W2 and W3, Figure 3A is a view showing the arrangement of the lens elements in an f lens of a third embodiment in accordance with the first aspect of the present invention, Figure 3B is a graph showing the lateral color aberration of the f lens shown in Figure 3A for three wavelengths W1, W2 and W3, Figure 3C is a graph showing the astigmatic field curves (curvature of field) of the f lens shown in Figure 3A for the three wavelengths W1, W2 and W3, Figure 3D is a graph showing the linearity (f characteristic) of the f lens shown in Figure 3A for the three wavelengths W1, W2 and W3, Figure 4A is a view showing the arrangement of the lens elements in an f lens of a fourth embodiment in accordance with the first aspect of the present invention, Figure 4B is a graph showing the lateral color aberration of the f lens shown in Figure 4A for three wavelengths W1, W2 and W3, Figure 4C is a graph showing the astigmatic field curves (curvature of field) of the f lens shown in Figure 4A for the three wavelengths W1, W2 and W3, Figure 4D is a graph showing the linearity (f characteristic) of the f lens shown in Figure 4A for the three wavelengths W1, W2 and W3, Figure 5A is a view showing the arrangement of the lens elements in an f lens of a fifth embodiment in accordance with the first aspect of the present invention, Figure 5B is a graph showing the lateral color aberration of the f lens shown in Figure 5A for three wavelengths W1, W2 and W3, Figure 5C is a graph showing the astigmatic field curves (curvature of field) of the f lens shown in Figure 5A for the three wavelengths W1, W2 and W3, Figure 5D is a graph showing the linearity (f characteristic) of the f lens shown in Figure 5A for the three wavelengths W1, W2 and W3, Figure 6A is a view showing the arrangement of the lens elements in an f lens of a sixth embodiment in accordance with the first aspect of the present invention, Figure 6B is a graph showing the lateral color aberration of the f lens shown in Figure 6A for three wavelengths W1, W2 and W3, Figure 6C is a graph showing the astigmatic field curves (curvature of field) of the f lens shown in Figure 6A for the three wavelengths W1, W2 and W3, Figure 6D is a graph showing the linearity (f characteristic) of the f lens shown in Figure 6A for the three wavelengths W1, W2 and W3, Figure 7A is a view showing the arrangement of the lens elements in an f lens of a seventh embodiment in accordance with the first aspect of the present invention, Figure 7B is a graph showing the lateral color aberration of the f lens shown in Figure 7A for three wavelengths W1, W2 and W3, Figure 7C is a graph showing the astigmatic field curves (curvature of field) of the f lens shown in Figure 7A for the three wavelengths W1, W2 and W3, Figure 7D is a graph showing the linearity (f characteristic) of the f lens shown in Figure 7A for the three wavelengths W1, W2 and W3, Figure 8A is a view showing the arrangement of the lens elements in an f lens of an eighth embodiment in accordance with the first aspect of the present invention, Figure 8B is a graph showing the lateral color aberration of the f lens shown in Figure 8A for three wavelengths W1, W2 and W3, Figure 8C is a graph showing the astigmatic field curves (curvature of field) of the f lens shown in Figure 8A for the three wavelengths W1, W2 and W3, Figure 8D is a graph showing the linearity (f characteristic) of the f lens shown in Figure 8A for the three wavelengths W1, W2 and W3, Figure 9A is a view showing the arrangement of the lens elements in an f lens of a ninth embodiment in accordance with the first aspect of the present invention, Figure 9B is a graph showing the lateral color aberration of the f lens shown in Figure 9A for three wavelengths W1, W2 and W3, Figure 9C is a graph showing the astigmatic field curves (curvature of field) of the f lens shown in Figure 9A for the three wavelengths W1, W2 and W3, Figure 9D is a graph showing the linearity (f characteristic) of the f lens shown in Figure 9A for the three wavelengths W1, W2 and W3, Figure 10A is a view showing the arrangement of the lens elements in an f lens of a tenth embodiment in accordance with the first aspect of the present invention, Figure 10B is a graph showing the lateral color aberration of the f lens shown in Figure 10A for three wavelengths W1, W2 and W3, Figure 10C is a graph showing the astigmatic field curves (curvature of field) of the f lens shown in Figure 10A for the three wavelengths W1, W2 and W3, Figure 10D is a graph showing the linearity (f characteristic) of the f lens shown in Figure 10A for the three wavelengths W1, W2 and W3, Figure 11A is a view showing the arrangement of the lens elements in an f lens of an eleventh embodiment in accordance with the first aspect of the present invention, Figure 11B is a graph showing the lateral color aberration of the f lens shown in Figure 11A for three wavelengths W1, W2 and W3, Figure 11C is a graph showing the astigmatic field curves (curvature of field) of the f lens shown in Figure 11A for the three wavelengths W1, W2 and W3, Figure 11D is a graph showing the linearity (f characteristic) of the f lens shown in Figure 11A for the three wavelengths W1, W2 and W3, Figure 12A is a view showing the arrangement of the lens elements in an f lens of an twelfth embodiment in accordance with the first aspect of the present invention, Figure 12B is a graph showing the lateral color aberration of the f lens shown in Figure 12A for three wavelengths Wl, W2 and W3, Figure 12C is a graph showing the astigmatic field curves (curvature of field) of the f lens shown in Figure 12A for the three wavelengths W1, W2 and W3, Figure 12D is a graph showing the linearity (f characteristic) of the f lens shown in Figure 12A for the three wavelengths W1, W2 and W3, Figure 13A is a view showing the arrangement of the lens elements in an f lens of an thirteenth embodiment in accordance with the first aspect of the present invention, Figure 13B is a graph showing the lateral color aberration of the f lens shown in Figure 13A for three wavelengths W1, W2 and W3, Figure 13C is a graph showing the astigmatic field curves (curvature of field) of the f lens shown in Figure 13A for the three wavelengths W1, W2 and W3, and Figure 13D is a graph showing the linearity (f characteristic) of the f lens shown in Figure 13A for the three wavelengths W1, W2 and W3. DESCRIPTION OF THE PREFERRED EMBODIMENTS f lenses of thirteen embodiments in accordance with the present invention will be described with reference to Figures 1 (1A to 1D) to 13 (13A to 13D) and tables 1 to 13, hereinbelow. Figures 1A to 13A respectively show the arrangements of the lens elements of the f lenses of first to thirteenth embodiments, and Figures 1B, 1C and 1D to 13B, 13C and 13D respectively show the lateral color aberration, the astigmatic field curves (curvature of field) and the linearity (f characteristic) of the f lenses of the first to thirteenth embodiments when three laser beams having different wavelengths W1, W2 and W3 used in a scanning optical system enter the respective f lenses. The values of the wavelengths W1, W2 and W3 are shown on the top of the tables corresponding to the f lenses. Each f lens comprises a first lens group which consists of a single first lens L1 having a negative or positive refracting power, and a second lens group which consists of a second lens L2 having a positive refracting power and a third lens L3 having a negative refracting power. The second and third lenses L2 and L3 are cemented together and the second lens group has a positive refracting power as a whole. The first and second lens groups are arranged in this order from the entrance pupil (deflecting surface) side and satisfies the following formulae (1) to (4), 12≦ν d2 -ν d3 -0.035≦Σ( i /ν di )≦0.005 0.7≦ 23 ≦2.0 0.03≦d 12 ≦0.20 wherein  i represents the power of i-th lens as numbered from the entrance pupil side standardized on the basis of the power of the whole system,  23 represents the power of the second lens group standardized on the basis of the power of the whole system, ν di represents the Abbe's number for the sodium d-line of i-th lens as numbered from the entrance pupil side and d 12 represents the air separation between the first and second lens groups standardized on the basis of the focal length of the whole system. In the first embodiment shown in Figure 1A, the factors limited by the formulae (2) to (4) are set substantially at the middle of the ranges defined by the formulae. The radii of curvature R of the refracting surfaces, the axial surface separations d, and the refractive indexes n d and the Abbe's numbers ν d of the lenses for the sodium d-line in the f lens of this embodiment are shown in table 1. The radii of curvature R of the refracting surfaces and the axial surface separations d are in the values standardized on the basis of the focal length of the whole system. (the same for all the embodiments including the embodiments in accordance with the second and third aspects of the present invention described later) The numbers in the leftmost column of table 1 designate the numbers of the symbols R, d, n d and ν d as numbered from the entrance pupil (deflecting surface) side. (the same for all the tables in this specification) wavelength W1;810nm, W2;750nm, W3;680nm, angle of view 62.5° i Ri di n d ν d OBJ - infinity 0 infinity 0.10391 1 -1.76735 0.02474 n d1 1.48749 ν d1 70.4 2 18.24431 0.07422 3 2.39800 0.09896 n d2 1.63930 ν d2 45.1 4 -0.21884 0.02474 n d3 1.80518 ν d3 25.5 5 -0.46351 1.03924 surface scanned infinity - ν d2 -ν d3 = 19.6 lateral color aberration 0.049x10 -3 Σ(i/ν di ) = -0.007 curvature of field 0.016  23 =1.24 f characteristic 1.16% d 12 =0.07422 Figures 1B to 1D respectively show the lateral color aberration, the astigmatic field curves (curvature of field) in the scanning direction and the linearity (f characteristic) of the f lens of the first embodiment when laser beams having wavelengths of 810nm (W1), 750nm (W2) and 680nm (W3) enter the f lens. In Figures 1C and 1D, the astigmatic field curves and the linearity for the wavelengths W1, W2 and W3 are respectively shown by the solid line, the fine broken line and the rough broken line, but when one or both of the broken lines coincide with the solid line, the broken line is not shown. (the same for all the embodiments including the embodiments in accordance with the second and third aspects of the present invention described later) As can be understood from Figures 1B to 1D, in the f lens of this embodiment, the lateral color aberration is well corrected for the respective wavelengths in spite of a simple structure consisting of only three lens elements in two lens groups. In the second embodiment shown in Figure 2A, the difference between the Abbe's numbers of the second lens L2 and the third lens L3 is set to a value close to the lower limit of the range defined by the formula (1). The radii of curvature R of the refracting surfaces, the axial surface separations d, and the refractive indexes n d and the Abbe's numbers ν d of the lenses for the sodium d-line in the f lens of this embodiment are shown in table 2. wavelength W1;810nm, W2;750nm, W3;680nm, angle of view 62.5° i Ri di n d ν d OBJ - infinity 0 infinity 0.10390 1 -1.19973 0.02474 n d1 1.80518 ν d1 25.5 2 7.58872 0.05332 3 2.00978 0.09895 n d2 1.72342 ν d2 38.0 4 -0.17854 0.02474 n d3 1.80518 ν d3 25.5 5 -0.46377 1.08380 surface scanned infinity - ν d2 -ν d3 = 12.5 lateral color aberration 0.039x10 -3 Σ(i/ν di ) = -0.020 curvature of field 0.014  23 =1.61 f characteristic 2.39% d 12 =0.05332 Figures 2B to 2D respectively show the lateral color aberration, the astigmatic field curves in the scanning direction and the linearity of the f lens of the second embodiment when laser beams having wavelengths of 810nm (W1), 750nm (W2) and 680nm (W3) enter the f lens. As can be understood from Figures 2B to 2D, in the f lens of this embodiment, the lateral color aberration is well corrected for the respective wavelengths in spite of a simple structure consisting of only three lens elements in two lens groups. In the third embodiment shown in Figure 3A, the surface separation d 12 between the first and second lens groups is set to a value close to the lower limit of the range defined by the formula (4). The radii of curvature R of the refracting surfaces, the axial surface separations d, and the refractive indexes n d and the Abbe's numbers ν d of the lenses for the sodium d-line in the f lens of this embodiment are shown in table 3. wavelength W1;810mn, W2;750nm, W3;680nm, angle of view 62.5° i Ri di n d ν d OBJ - infinity 0 infinity 0.10393 1 -14.8476 0.02475 n d1 1.80518 ν d1 25.5 2 0.73495 0.03228 3 0.94676 0.09898 n d2 1.62012 ν d2 49.8 4 -0.17322 0.02475 n d3 1.64769 ν d3 33.8 5 -0.42690 1.07406 surface scanned infinity ν d2 - ν d3 = 16.0 lateral color aberration 0.074x10 -3 Σ(i/ν di ) = -0.025 curvature in field 0.016  23 =1.94 f characteristic 3.04% d 12 =0.03228 Figures 3B to 3D respectively show the lateral color aberration, the astigmatic field curves in the scanning direction and the linearity of the f lens of the third embodiment when laser beams having wavelengths of 810nm (W1), 750nm (W2) and 680nm (W3) enter the f lens. As can be understood from Figures 3B to 3D, in the f lens of this embodiment, the lateral color aberration is well corrected for the respective wavelengths in spite of a simple structure consisting of only three lens elements in two lens groups. In the fourth embodiment shown in Figure 4A, the first lens L1 has a negative refracting power and the surface separation d 12 between the first and second lens groups is set to a relatively large value in the range defined by the formula (4). The radii of curvature R of the refracting surfaces, the axial surface separations d, and the refractive indexes n d and the Abbe's numbers ν d of the lenses for the sodium d-line in the f lens of this embodiment are shown in table 4. wavelength W1;810nm, W2;750nm, W3;680nm, angle of view 62.5° i Ri di n d ν d OBJ - infinity 0 infinity 0.10390 1 -0.45615 0.02474 n d1 1.51680 ν d1 64.2 2 -0.48189 0.14842 3 8.15793 0.11379 n d2 1.66672 ν d2 48.3 4 -0.28119 0.02474 n d3 1.80518 ν d3 25.5 5 -0.56734 1.02736 surface scanned infinity - ν d2 - ν d3 = 22.8 lateral color aberration 0.055x10 -3 Σ(i/ν di ) =-0.004 curvature of field 0.015  23 =1.01 f characteristic 1.03% d 12 =0.14842 Figures 4B to 4D respectively show the lateral color aberration, the astigmatic field curves in the scanning direction and the linearity of the f lens of the fourth embodiment when laser beams having wavelengths of 810nm (W1), 750nm (W2) and 680nm (W3) enter the f lens. As can be understood from Figures 4B to 4D, in the f lens of this embodiment, the lateral color aberration is well corrected for the respective wavelengths in spite of a simple structure consisting of only three lens elements in two lens groups. In the fifth embodiment shown in Figure 5A, the first lens L1 has a positive refracting power. The radii of curvature R of the refracting surfaces, the axial surface separations d, and the refractive indexes n d and the Abbe's numbers ν d of the lenses for the sodium d-line in the f lens of this embodiment are shown in table 5. wavelength W1;810nm, W2;750nm, W3;680nm, angle of view 62.5° i Ri di n d ν d OBJ - infinity 0 infinity 0.10390 1 -0.14842 0.02474 n d1 1.51680 ν d1 64.2 2 -0.14564 0.13018 3 1.25361 0.09895 n d2 1.53172 ν d2 48.8 4 -0.30864 0.02474 n d3 1.80518 ν d3 25.5 5 -0.67737 0.97906 surface scanned infinity ν d2 - ν d3 = 23.3 lateral color aberration 0.034x10 -3 Σ(i/ν di ) = -0.006 curvature of field 0.011  23 =0.73 f characteristic 2.52% d 12 =0.13018 Figures 5B to 5D respectively show the lateral color aberration, the astigmatic field curves in the scanning direction and the linearity of the f lens of the fifth embodiment when laser beams having wavelengths of 810nm (W1), 750nm (W2) and 680nm (W3) enter the f lens. As can be understood from Figures 5B to 5D, in the f lens of this embodiment, the lateral color aberration is well corrected for the respective wavelengths in spite of a simple structure consisting of only three lens elements in two lens groups. In the sixth embodiment shown in Figure 6A, the factors limited by the formulae (2) to (4) are set substantially at the middle of the ranges defined by the formulae. The radii of curvature R of the refracting surfaces, the axial surface separations d, and the refractive indexes n d and the Abbe's numbers ν d of the lenses for the sodium d-line in the f lens of this embodiment are shown in table 6. wavelength W1;870nm, W2;810nm, W3;750nm, angle of view 62.5° i Ri di n d ν d OBJ - infinity 0 infinity 0.10390 1 -0.69730 0.02474 n d1 1.51680 ν d1 64.2 2 -4.91198 0.07766 3 3.12259 0.09895 n d2 1.65844 ν d2 50.9 4 -0.18906 0.02474 n d3 1.72825 ν d3 28.3 5 -0.44235 1.09609 surface scanned infinity - ν d2 -ν d3 = 22.6 lateral color aberration 0.041x10 -3 Σ(i/ν di ) = -0.012 curvature of field 0.015  23 = 1.48 f characteristic 1.88% d 12 =0.07766 Figures 6B to 6D respectively show the lateral color aberration, the astigmatic field curves in the scanning direction and the linearity of the f lens of the sixth embodiment when laser beams having wavelengths of 870nm (W1), 810nm (W2) and 750nm (W3) enter the f lens. As can be understood from Figures 6B to 6D, in the f lens of this embodiment, the lateral color aberration is well corrected for the respective wavelengths in spite of a simple structure consisting of only three lens elements in two lens groups. In the seventh embodiment shown in Figure 7A, the factors limited by the formulae (2) to (4) are set substantially at the middle of the ranges defined by the formulae. The radii of curvature R of the refracting surfaces, the axial surface separations d, and the refractive indexes n d and the Abbe's numbers ν d of the lenses for the sodium d-line in the f lens of this embodiment are shown in table 7. wavelength W1;890nm, W2;830nm, W3;780nm, angle of view 62.5° i Ri di n d ν d OBJ - infinity 0 infinity 0.10390 1 -0.57722 0.02474 n d1 1.56732 ν d1 42.8 2 -1.68735 0.08847 3 4.06146 0.09895 n d2 1.62299 ν d2 58.1 4 -0.20052 0.02474 n d3 1.72825 ν d3 28.3 5 -0.39721 1.11410 surface scanned infinity - ν d2 - ν d3 = 29.8 lateral color aberration 0.029x10 -3 Σ(i/ν di ) = -0.019 curvature of field 0.013  23 = 1.46 f characteristic 1.75% d 12 =0.08847 Figures 7B to 7D respectively show the lateral color aberration, the astigmatic field curves in the scanning direction and the linearity of the f lens of the seventh embodiment when laser beams having wavelengths of 890nm (W1), 830nm (W2) and 780nm (W3) enter the f lens. As can be understood from Figures 7B to 7D, in the f lens of this embodiment, the lateral color aberration is well corrected for the respective wavelengths in spite of a simple structure consisting of only three lens elements in two lens groups. In the eighth embodiment shown in Figure 8A, the first lens L1 has a negative refracting power and the difference between the longest and the shortest of the wavelengths W1 to W3 of the laser beams entering the f lens is enlarged. The radii of curvature R of the refracting surfaces, the axial surface separations d, and the refractive indexes n d and the Abbe's numbers ν d of the lenses for the sodium d-line in the f lens of this embodiment are shown in table 8. wavelength W1;680nm, W2;532nm, W3;473nm, angle of view 55.7° i Ri di n d ν d OBJ - infinity 0 infinity 0.14278 1 -1.61585 0.02380 n d1 1.60000 ν d1 42.5 2 1.80042 0.04427 3 1.08641 0.09519 n d2 1.49700 ν d2 81.6 4 -0.18913 0.01904 n d3 1.64844 ν d3 50.9 5 -0.32890 1.06515 surface scanned infinity - ν d2 -ν d3 = 30.7 lateral color aberration 0.071x10 -3 Σ(i/ν di ) = -0.007 curvature of field 0.002  23 = 1.58 f characteristic 1.44% d 12 =0.04427 Figures 8B to 8D respectively show the lateral color aberration, the astigmatic field curves in the scanning direction and the linearity of the f lens of the eighth embodiment when laser beams having wavelengths of 680nm (W1), 532nm (W2) and 473nm (W3) enter the f lens. As can be understood from Figures 8B to 8D, in the f lens of this embodiment, the lateral color aberration is well corrected for the respective wavelengths in spite of a simple structure consisting of only three lens elements in two lens groups. In the ninth embodiment shown in Figure 9A, the first lens L1 has a positive refracting power and the difference between the longest and the shortest of the wavelengths W1 to W3 of the laser beams entering the f lens is enlarged. The radii of curvature R of the refracting surfaces, the axial surface separations d, and the refractive indexes n d and the Abbe's numbers ν d of the lenses for the sodium d-line in the f lens of this embodiment are shown in table 9. wavelength W1;680nm, W2;532nm, W3;473nm, angle of view 55.7° i Ri di n d ν d OBJ - infinity 0 infinity 0.09518 1 -0.16194 0.01666 n d1 1.65844 ν d1 50.9 2 -0.16505 0.09968 3 1.37817 0.07138 n d2 1.49700 ν d2 81.6 4 -0.21639 0.02855 n d3 1.67003 ν d3 47.2 5 -0.50462 1.01642 surface scanned infinity - ν d2 -ν d3 = 34.4 lateral color aberration 0.071x10 -3 Σ(i/ν di ) = -0.002 curvature of field 0.011  23 = 0.89 f characteristic 0.69% d12=0.09968 Figures 9B to 9D respectively show the lateral color aberration, the astigmatic field curves in the scanning direction and the linearity of the f lens of the ninth embodiment when laser beams having wavelengths of 680nm (W1), 532nm (W2) and 473nm (W3) enter the f lens. As can be understood from Figures 9B to 9D, in the f lens of this embodiment, the lateral color aberration is well corrected for the respective wavelengths in spite of a simple structure consisting of only three lens elements in two lens groups. As can be understood from the description above, in the f lens in accordance with the first aspect of the present invention, the color aberration can be corrected with a high accuracy over a wide wavelength range, and accordingly, the error in superimposing the beam spots can be reduced in the scanning optical system or the like using a plurality of laser beams having different wavelengths. Further since being simple in structure, the f lens in accordance with the first aspect of the present invention can be manufactured at low cost. The f lens in accordance with the first aspect of the present invention need not be limited to those described above but may be variously modified so long as the formulae (1) to (4) are satisfied. The f lenses of tenth to thirteenth embodiments shown in Figures 10A to 13A comprise an aspheric lense. In the tenth embodiment shown in Figure 10A, the first surface of the first lens L1 (the entrance surface) is an aspheric surface as shown in table 10 and the factors limited by the formulae (2) to (4) are set substantially at the middle of the ranges defined by the formulae. The radii of curvature R of the refracting surfaces, the axial surface separations d, and the refractive indexes n d and the Abbe's numbers ν d of the lenses for the sodium d-line in the f lens of this embodiment are shown in table 10. In table 10, the value designated by represents the radius of curvature on the optical axis (at the vertex of the aspheric surface) and means that the aspheric surface has a shape defined by the following formula (13). (the same in tables 11 to 13, 21 to 24, 34 and 35) x=ch 2 /[1+{1-(1+k)c 2 h 2 } 1/2 ]+a 1 h 4 +a 2 h 6 +a 3 h 8 +a 4 h 10 wherein x represents the length of a perpendicular line between a point on an aspheric plane at a height of h above the optical axis and a plane tangent to the aspheric plane (a plane perpendicular to the optical axis) (as standardized on the basis of the focal length of the whole system), h represents the height above the optical axis (as standardized on the basis of the focal length of the whole system), c represents the curvature at the vertex of the aspheric plane (the reciprocal of the radius of curvature =1/R), k represents a conic constant, and a 1 to a 4 respectively represents four-dimensional, dimensional, six-dimensional, eight-dimensional and aspheric coefficients. Figures 10B to 10D respectively show the lateral color aberration, the astigmatic field curves in the scanning direction and the linearity of the f lens of the tenth embodiment when laser beams having wavelengths of 810nm (W1), 750nm (W2) and 680nm (W3) enter the f lens. As can be understood from Figures 10B to 10D, in the f lens of this embodiment, the lateral color aberration is well corrected for the respective wavelengths in spite of a simple structure consisting of only three lens elements in two lens groups. In the eleventh embodiment shown in Figure 11A, the second surface of the first lens L1 (the exit surface) is an aspheric surface as shown in table 11 and the factors limited by the formulae (2) to (4) are set substantially at the middle of the ranges defined by the formulae. The radii of curvature R of the refracting surfaces, the axial surface separations d, and the refractive indexes n d and the Abbe's numbers ν d of the lenses for the sodium d-line in the f lens of this embodiment are shown in table 11. Figures 11B to 11D respectively show the lateral color aberration, the astigmatic field curves in the scanning direction and the linearity of the f lens of the eleventh embodiment when laser beams having wavelengths of 870nm (W1), 810nm (W2) and 750nm (W3) enter the f lens. As can be understood from Figures 11B to 11D, in the f lens of this embodiment, the lateral color aberration is well corrected for the respective wavelengths in spite of a simple structure consisting of only three lens elements in two lens groups. In the twelfth embodiment shown in Figure 12A, the first surface of the second lens L2 (the entrance surface) is an aspheric surface as shown in table 12, the first lens L1 has a positive refracting power and the difference between the longest and the shortest of the wavelengths W1 to W3 of the laser beams entering the f lens is enlarged. The radii of curvature R of the refracting surfaces, the axial surface separations d, and the refractive indexes n d and the Abbe's numbers ν d of the lenses for the sodium d-line in the f lens of this embodiment are shown in table 12. Figures 12B to 12D respectively show the lateral color aberration, the astigmatic field curves in the scanning direction and the linearity of the f lens of the twelfth embodiment when laser beams having wavelengths of 680nm (W1), 532nm (W2) and 473nm (W3) enter the f lens. As can be understood from Figures 12B to 12D, in the f lens of this embodiment, the lateral color aberration is well corrected for the respective wavelengths in spite of a simple structure consisting of only three lens elements in two lens groups. In the thirteenth embodiment shown in Figure 13A, the second surface of the third lens L3 (the exit surface) is an aspheric surface as shown in table 13 and the difference between the longest and the shortest of the wavelengths W1 to W3 of the laser beams entering the f lens is enlarged. The radii of curvature R of the refracting surfaces, the axial surface separations d, and the refractive indexes n d and the Abbe's numbers ν d of the lenses for the sodium d-line in the f lens of this embodiment are shown in table 13. Figures 13B to 13D respectively show the lateral color aberration, the astigmatic field curves in the scanning direction and the linearity of the f lens of the thirteenth embodiment when laser beams having wavelengths of 680nm (W1), 532nm (W2) and 473nm (W3) enter the f lens. As can be understood from Figures 13B to 13D, in the f lens of this embodiment, the lateral color aberration is well corrected for the respective wavelengths in spite of a simple structure consisting of only three lens elements in two lens groups.

Claims: An f lens comprising a first lens group which consists of a first lens (L1) having a negative or positive refracting power, and a second lens group which consists of a second lens (L2) having a positive refracting power and a third lens (L3) having a negative refracting power, the second and third lenses (L2,L3) being cemented together, the second lens group (L2,L3) having a positive refracting power as a whole, and the first and second lens groups being arranged in this order from the entrance pupil side and satisfying the following formulae (1) to (4), 12≦ν d2 -ν d3 -0.035≦Σ( i /ν di )≦0.005 0.7≦ 23 ≦2.0 0.03≦d 12 ≦0.20 wherein  i represents the power of i-th lens as numbered from the entrance pupil side standardized on the basis of the power of the whole system,  23 represents the power of the second lens group (L2,L3) standardized on the basis of the power of the whole system, ν di represents the Abbe's number for the sodium d-line of i-th lens as numbered from the entrance pupil side and d 12 represents the surface separation between the first and second lens groups standardized on the basis of the focal length of the whole system.

Inventors: NOGUCHI MASARU, SHIMADA KATSUMI, TSUJITA KAZUHIRO, NOGUCHI, MASARU, SHIMADA, KATSUMI, TSUJITA, KAZUHIRO, NOGUCHI, MASARU, C/O FUJI PHOTO FILM CO. LTD., SHIMADA, KATSUMI, C/O FUJI PHOTO FILM CO. LTD., TSUJITA, KAZUHIRO, C/O FUJI PHOTO FILM CO. LTD.
Applicants: FUJI PHOTO FILM CO LTD, FUJI PHOTO FILM CO., LTD.
Application year: 1994
Main classification: G02B13
IPC codes: G02B13